🏹 Parabola Calculator
A parabola is the U-shaped curve of every quadratic function — and of satellite dishes, headlight reflectors and thrown balls. Its key landmarks are the vertex (the turning point), the focus and directrix (the pair that defines the curve), and the axis of symmetry through them.
Enter your equation in vertex form y = a(x − h)² + k or standard form y = ax² + bx + c, and this calculator finds every landmark, both intercepts and the roots, and draws the parabola with its focus and directrix.
How to use this calculator
Vertex form shows the landmarks directly; standard form converts with h = −b/(2a) and k = c − b²/(4a).
- Choose the form of the equation you have.
- Enter a, and then h and k (vertex form) or b and c (standard form).
- Press Calculate.
- Read the vertex, focus, directrix and intercepts, and see them on the graph.
Parabola Worked Example
Analyse y = (x − 1)² − 4. It is already in vertex form with a = 1, h = 1, k = −4, so the vertex is (1, −4) and the axis is x = 1. The focal distance is p = 1/(4a) = 0.25, putting the focus at (1, −3.75) and the directrix at y = −4.25. Setting y = 0 gives (x − 1)² = 4, so the roots are x = 3 and x = −1; the y-intercept is (0, −3). These are the defaults above.
Frequently asked questions
What is the formula for the focus and directrix?
For y = a(x − h)² + k, the focal distance is p = 1/(4a). The focus is (h, k + p) and the directrix is the line y = k − p. Every point on a parabola is exactly as far from the focus as from the directrix — that is the definition of the curve.
How do I convert standard form to vertex form?
Completing the square: h = −b/(2a) and k = c − b²/(4a) (equivalently, k is the y-value at x = h). For example y = x² − 2x − 3 has h = 1 and k = −3 − 1 = −4, so it is y = (x − 1)² − 4. This calculator converts in both directions.
What does the sign and size of "a" do?
The sign decides the direction: positive a opens upward, negative opens downward. The size decides the width: larger |a| makes a narrower, steeper parabola; |a| below 1 makes a wider, flatter one. It also moves the focus — p = 1/(4a) shrinks as |a| grows.
When does a parabola have no x-intercepts?
When the vertex is on the opposite side of the x-axis from the opening direction — an upward parabola whose vertex sits above the axis (k > 0 with a > 0) never reaches y = 0. Algebraically, the discriminant of the quadratic is negative, so the roots are complex.